Geometries and Electronic States of Divacancy Defect in Finite …downloads.hindawi.com/journals/jchem/2017/8491264.pdf · B3LYP/STO-3G ΔEof 585defects ΔEof 555777defects ΔEbetween

Research ArticleGeometries and Electronic States of Divacancy Defect inFinite-Size Hexagonal Graphene Flakes

Lili Liu1 and Shimou Chen2

1Department of Chemistry, School of Science, Beijing Technology and Business University, Beijing 100048, China2Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences,Beijing 100190, China

Correspondence should be addressed to Shimou Chen; [email protected]

Received 14 December 2016; Accepted 4 January 2017; Published 29 January 2017

Academic Editor: Gang Feng

Copyright © 2017 Lili Liu and Shimou Chen. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The geometries and electronic properties of divacancies with two kinds of structures were investigated by the first-principles (U)B3LYP/STO-3G and self-consistent-charge density-functional tight-binding (SCC-DFTB) method. Different from the reportedunderstanding of these properties of divacancy in graphene and carbon nanotubes, it was found that the ground state of thedivacancy with 585 configurations is closed shell singlet state and much more stable than the 555777 configurations in the smallergraphene flakes, which is preferred to triplet state. But when the sizes of the graphene become larger, the 555777 defects will bemore stable. In addition, the spin density properties of the both configurations are studied in this paper.

1. Introduction

Graphene, a single carbon plane arranged on a honeycomblattice, has attracted immense investigation since its dis-covery in 2004 [1]. Lattice imperfections are introducedinto graphene unavoidably during graphene growth or whenirradiating a graphene sheet using high-energy particles[2–5]. These structural defects are known to significantlyaffect electronic and chemical properties [6, 7]. In particular,the presence of defects can dramatically change the chargetransport and magnetic properties of graphene due to dis-order and localization effects, which are of important fortheir applications in molecular electronics. A comprehensiveunderstanding of graphene defects is thus of critical impor-tance.

Divacancy defect in graphene can be obtained eitherby the coalescence of two monovacancies or by remov-ing two neighboring atoms. Although monovacancies hadbeen widely studied [8–12], the divacancy as well as morelarger vacancies could be frequently created in a physical orchemical treatment of graphene. For example, transmissionelectron microscopy experiments indicated that multivacan-cies, rather than monovacancies, more easily occur under

moderate irradiation conditions [13].Theoretical calculationsat different levels of theory also point out that formationenergies of divacancies in carbon nanotubes and graphene aremuch lower than monovacancy formation energies [14–18].

Typically, no dangling bond is present in a fully recon-structed divacancy so that two pentagons and one octagon(585 defects) appear, but the 585 defects are not the onlypossible way for a graphene lattice to accommodate twomissing atoms [19]. Lee et al. found that the rotation of oneof the bonds in the octagon of the 585 defects transformsthe defect into an arrangement of three pentagons and threeheptagons (555777 defects) [16, 20], which is also observedby the experiment study [21]. Even though high-resolutiontransmission electron microprobe methods enable an in situmeasurement on the stability and migration of divacancy atatomic scale, the microscopic identification of the defects cannot be an insight into the electronic properties of the defectedgraphene. In addition, as the edge (zigzag or armchair)plays a key role in the electronic and magnetic propertiesof graphene, as far as we know, there is still no report oninvestigating the influence on the electronic state of divacancyin graphene by edge effect.

HindawiJournal of ChemistryVolume 2017, Article ID 8491264, 7 pageshttps://doi.org/10.1155/2017/8491264

https://doi.org/10.1155/2017/8491264

2 Journal of Chemistry

In this work, the geometries and electronic states oftwo preferable types of divacancy defects in finite-sizehexagonal graphene flakes (HGFs) were calculated by self-consistent-charge density-functional tight-binding (SCC-DFTB) method for the first time, which are three pentagonsand three heptagons composed (555777) divacancy defectstructure and two pentagons side with octagon composed(585) structures, respectively. The formation energy, elec-tronic state, and spin density distribution of different isomerswere compared, and the edge effects on the properties of thesedifferent type divacancies were also studied.

2. Methodology and Model Structures

To calculate the geometries and electronic states of diva-cancy in graphene, we employed the computationally moreeconomic self-consistent-charge density-functional tight-binding (SCC-DFTB) method in addition to first-principles(U) B3LYP/STO-3G level of theory to do the geometry opti-mization and the different spin states calculations. The SCC-DFTB calculations were carried out with DFTB+, a programpackage developed by Aradi, Hourahine, and Frauenheim,implementing the original DFTB algorithms using sparse-matrix techniques [22–26]. The electronic temperature 𝑇efor the open shell state calculation was chosen to be 1000K,and the spin-polarized version of DFTB was employed forthe high spin state calculation. For numerical vibrationalfrequencies calculation, we used GAUSSIAN 03 package with“external” keyword, calling SCC-DFTB 𝑇e = 1000K external= “SCCDFTB” [27].

In this work, two series of typical divacancy defectsin different sizes of zigzag- and armchair-edges hexagonalgraphene flakes were investigated (Figure 1). One structureis three pentagons and three heptagons composed structures(555777) divacancy defect (highlighted by black color inFigure 1(a)), and the other type is two pentagons’ side withan octagon composed (585) configuration (highlighted byblack color in Figure 1(b)). There are different sizes of theflakes with divacancy for the same defect structure: fromZ1 to Z12 for zigzag HGFs and from A1 to A6 for armchairHGFs (the nomination was shown in Figure 1). In addition,the small examples of the two defects are shown in Figures1(b) and 1(c); we call these two types defects 585 and 555777,respectively. All these structures are minima, which haveperformed numerical vibrational frequencies calculation.

3. Results and Discussion

3.1. The Electronic State Properties of the Different DivacancyIsomers. In order to study the electronic characteristic ofthe divacancy defects, the energy differences between thedifferent electronic states have been calculated on the twotypes of defects using both B3LYP/STO-3G and DFTBmethod and the results are shown in Figure 2. Figure 2(a)shows the energy differences of the same isomers with thedifferent edge shape using B3LYP method, and Figure 2(b)shows the results using DFTB method; because the B3LYPis a very expensive calculationmethod for big size system, we

only performed the finite-sized graphene flake. Both results ofthe energy differences with the two methods can be derivedthatDFTBmethodproduced the same tendency of the energydifferences with B3LYPmethod, and it is reliable and efficientin the really large carbon system.

Based on the comprehensive results of DFTB calculationin Figure 2, all theΔ𝐸 values (Δ𝐸 = 𝐸triplet−𝐸singlet) of 555777defects are negative except for Z1 in zigzag-edge grapheneflakes and all the values of 585 defects are positive exceptfor Z11 and Z12. It stands for the notion that for the 555777structures the ground states are more preferably triplet, whilethe 585 structures are preferred to be closed shell singletwithin the zigzag HGFs. But for the divacancy in armchairHGFs, the energy difference curve is more flat, especiallyfor the 555777 defects, the energies of singlet and tripletstructures are very close to each other, all the ground states of585 structures are closed shell states, and the 555777 defectsare triplet states.

3.2. The Stability of 55577 and 585 Divacancies. The stabilityof the two different configurations of the divacancies in thesame size and edge graphene also has been calculated withboth DFT and DFTB methods. The results are shown inFigure 3. From the figure, it is easy to see that, for themediumsizes HGFs, the divacancy energy differences of the two levelsof theories are very similar, which also proved that the DFTBis reliable for such system.

In addition, when the Δ𝐸 values, the energy differencesbetween ground state energies of 585 defects (singlet state)and 555777 defects (triplet state), are positive, it means the555777 is more stable, while when the value is negative, itmeans the 585 structures are more stable. From Figure 3, onecan conclude that the stability has a very close relationshipwith both the HGFs edge shape and the defect structure. Inarmchair-edge HGFs, when the flakes size is smaller than A5(A5 means the structure with a 18.5 A distance between thecenters and the hexagon corner), the Δ𝐸 values are negative,and the 585 structure defects will be more stable than 555777defects. That is to say, from A5 to bigger HGFs, the open-shelled triplet 555777 divacancy defects changed to be morestable. But for zigzag-edge HGFs, this stability changed fromZ8 (Z8 means the structure with a 21.0 A distance betweenthe center and the hexagon corner), and from that structure,the 585 divacancies changed to be more stable than 555777structures. So the stability of these two popular divacanciesare highly depended on the structures themselves and alsodetermined by the HGFs edges.

3.3.The Formation Energies of the Two Divacancy Isomers. Inthe former theoretical studies of the divacancy on graphene,it is reported that the 585 defects are less stable than 555777defects about 0.8 eV in graphene [16, 17]. But in the nan-otubes, it had been predicted that the formation energies fordivacancies in armchair nanotubes are higher than in zigzagnanotubes. In addition, for the 585 divacancies in CNTs, theformation energies of divacancies strongly depend on theorientation of the divacancy with respect to the tube axis[27]. Lee et al. carried out the first-principles calculations and

Journal of Chemistry 3

A1

A3

A4

A5

A2

A6

Z1 Z7 Z8 Z9 Z10

Z11

Z12Z6Z5Z4Z3Z2

(a)

Z1 Z3Z2

A2

A1

(b)

Z1 Z3Z2A2

A1

(c)

Figure 1: (a) The model structures of the two types divacancies, black bond stands for the zigzag-edge hexagonal graphene flakes and redbonds stand for the armchair ones. (b) Example for the small sizes of the zigzag (Z1, Z2, Z3) and armchair (A1, A2) hexagonal flakes with 585defects in the center. (c) Example for the small sizes of the zigzag and armchair hexagonal flakes with 555777 defects in the center.

confirmed that the 555-777 defect is energetically muchmorestable in graphene [16]. In order to clarify the controversyresult on this issue, here, we performed DFTB calculation onthe formation energy of divacancy inHGFswith different sizeand edge of graphene.

The formation energy was calculated based on the follow-ing formula:

Δ𝐸 = 𝐸 (Z𝑛) − 𝐸 (Z

𝑛+2) + 2 ∗ 𝐸 (C atom) , (1)

where the symbol Z𝑛stands for the divacancy defects shown

in Figure 1, Z𝑛+2

stands for the HGFs without defects, andZ represents the zigzag-edged graphene flakes. The armchairdivacancy defects are calculated in the same way.

The results of our calculations are shown in Figure 4.The divacancy formation energies of the two isomers (555777and 585) in the same HGFs are increasing along with the

HGFs size increase. For the zigzag-edge graphene flakes, theformation energy of the 555777 defects are higher than 585defects in the size rang from Z1 to Z7, wherein, when the sizeofHGFs is equal to or bigger than Z8, the formation energy of585 becomemore higher, which have the same tendency withthe energy differences between these two isomers in the sameflakes as shown in Figure 2. For the armchair-edge HGFs, theresult of our calculation also shows the similar tendency onthe formation energy of the two isomers, but, in this case,the turning point is A5, the formation energy of 555777 ishigher than that of 585 when the size is smaller than A5 andthen becomes lower when the size of armchair-edge grapheneis the same or higher. Therefore, we obtained a general ruleon the formation energy of graphene flakes, and we foundthat the absolute value of the formation energy has thesame tendency with the energy differences between the two


−0.400

−0.200

0.000

0.200

0.400

0.600

0.800

1.000

1.200

ΔE of 585 defectsΔE of 555777 defects

ΔE between singlet and triplet states of zigzag graphene

2 4 60B3LYP/STO-3G


ΔE between singlet and triplet states of armchair graphene

−0.600−0.400−0.200

0.0000.2000.4000.6000.8001.0001.2001.4001.600

1 2 3 40B3LYP/STO-3G

(a)

ΔE between singlet and triplet states of zigzag graphene


−1.40−1.20−1.00−0.80−0.60−0.40−0.20

0.000.200.400.60

ΔE

(eV

)

1 2 3 4 5 6 7 8 9 10 11 12 130DFTB

ΔE between singlet and triplet states of armchair graphene


−1.40

−1.20

−1.00

−0.80

−0.60

−0.40

−0.20

0.00

0.20

ΔE

(eV

)

1 2 3 4 5 6 70DFTB

(b)

Figure 2:The energy differences between the singlet and triplet states of different types of divacancy defects with B3LYP/STO-3G and SDFTBmethods, respectively.

isomers in the same flakes, and the stability of the defectiveflakes is highly dependent on the size of the graphene flakesand the defect structures.

3.4. Spin Density Distribution of the Open-Shelled 555777Defects. According to the analysis results above, the groundstates of the two divacancy defects are totally different. Themost stable electronic state of the 585 divacancies is closeshell singlet without any unpaired electron, but for the 555777structures, the energetically stable state is open shell tripletstate, which has two unpaired electrons. The two unpairedelectrons are evenly localized on three carbons that aredirectly connected to the center carbon atom and have beenshared by both the pentagon and the heptagon. Some middlesize zigzag and armchair 555777 divacancy graphene defectsof the spin density plots are shown in Figure 5. It is clear that

the location of the unpaired electrons is within the isovalue= ±0.01. The electrons locations are all the same in the twodifferent edge HGFs and have no relationship with the sizeor the edge of the graphene flakes. From this calculation, wefirst found that the spin density of the 555777 divacancy onlydepends on the vacancy itself. The radical property carbonhas more relativities indicating that it is very important tounderstand the process of the divacancy defect absorption orself-healing reaction.

4. Conclusions

In conclusion, we have investigated the geometries and elec-tronic properties of divacancies with two kinds of structuresin the different edged hexagonal graphene flakes by the first-principles (U) B3LYP/STO-3G and self-consistent-charge


Energy difference between 585 and 555777 defects

ZigzagArmchair

ZigzagArmchair

−2.50

−2.00

−1.50

−1.00

−0.50

0.00

0.50

1.00

ΔE

(eV

)

ΔE = E585 − E555777

ΔE between 585 and 555777 by B3LYP/STO-3G ΔE between 585 and 555777 by DFTB

2 4 60 1 2 3 4 5 6 7 8 9 10 11 12 130

−2.50

−2.00

−1.50

−1.00

−0.50

0.00

0.50

1.00

ΔE

(eV

)Figure 3: The energy differences between different types of divacancy defects (Δ𝐸 = 𝐸

585− 𝐸555777) in the same graphene flakes with

B3LYP/STO-3G and DFTB methods.

Formation energy of divacancy in zigzag-edge graphene flakes

585 defects555777 defects

21.50

22.00

22.50

23.00

23.50

24.00

24.50

25.00

25.50

26.00

26.50

27.00

Form

atio

n en

ergy

(eV

)

1 2 3 4 5 6 7 8 9 10 11 12 130

Formation energy of divacancy in armchair-edge graphene flakes

585 defects555777 defects

21.50

22.00

22.50

23.00

23.50

24.00

24.50

25.00

25.50

26.00

26.50

27.00

Form

atio

n en

ergy

(eV

)

1 2 3 4 5 6 70

Figure 4: The formation energy of the two types of divacancy defects in zigzag- and armchair-edge HGFs by DFTB.

density-functional tight-binding (SCC-DFTB) method. TheDFTB reproduced the B3LYP results correspondingly andgave very good results even in the large system, which provedthat it is reliable and efficient method in the really largecarbon system. It is found that the ground state of thedivacancy with 585 configuration is closed shell singlet stateand much more stable than the 555777 configurations in thecertain small size range, which is preferred to triplet state.More importantly, different from the reported understandingof these properties of divacancy in graphene, we found that

the stability of these two kinds of divacancies is highlydependent on the structures themselves and also depends onthe HGFs edges characters. Through the investigation of thedifferent states energy comparison and formation energy, wefound that the stability of the two divacancies is influencednot only by the size of the graphene flakes but also by theedge type of graphene. In addition, we also calculated the spindensity of 555777 and found that the spin density distribu-tion of the 555777 divacancy only depends on the vacancyitself.


Z2 Z3 Z4 Z5

A2 A3 A4 A5

Figure 5: The spin density distributions of the 555777 divacancy defects in the two type edges of HGFs with the isovalue = ±0.01.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (nos. 21276257, 91534109, and 21503006),the “Strategic Priority Research Program” of the ChineseAcademy of Sciences (no. XDA09010103), National KeyProjects for Fundamental Research and Development ofChina (no. 2016YFB0100100), and the Research Foundationfor Youth Scholars of Beijing Technology and BusinessUniversity (no. QNJJ2014-14).

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Geometries and Electronic States of Divacancy Defect in Finite …downloads.hindawi.com/journals/jchem/2017/8491264.pdf · B3LYP/STO-3G ΔEof 585defects ΔEof 555777defects ΔEbetween